Lessons 7.1 Students learn about a function through trying to un-code a message ◦ They will notice that unambiguous encoding requires a function.

Lessons ◦ Students learn about the properties and geometric representation of a function. ◦ Students learn the difference between independent and dependent variables and construct and interpret simple graphs that related to real-world data. ◦ Students learn about function notation and its relationship to input and output variables.

Lessons ◦ Students learn several ways to define absolute value, and then construct and interpret graphs of absolute value functions. ◦ Students learn about squaring (parabolas) and square root functions and relate squaring a function to finding the area of a square.

LESSON 7.1 Functions To investigate the concept of function through secret codes.

In this lesson you will use a coding grid to write a coded message create and use a letter-shift code determine whether given relationships are functions

You have studied many relationships between variables. In this lesson you will learn about a special type of relationship called a function.

The letter A is coded into the letter Q Original input ABCDEFGHIJKLM Coded output QRSTUVWXYZABC Original input NOPQRSTUVRXYZ Coded output DEFGHIJKLMNOP The letter B is coded into the letter R The letter U is coded into the letter K This is an example of a letter-shift code. How would you use the code to write a message? Original Input Coded Output

Use the coding grid to write a two-word or three-word message. Exchange your coded message with a partner. Use this grid to decode each other’s messages. Original input ABCDEFGHIJKLM Coded output QRSTUVWXYZABC Original input NOPQRSTUVRXYZ Coded output DEFGHIJKLMNOP Original Input Coded Output

Create a new code by writing a rule that shifts letters a certain specified number of places. Put the code on a grid like the one shown on the last slides. Do not let your partner see the grid. Original input ABCDEFGHIJKLM Coded output Original input NOPQRSTUVRXYZ Coded output Use your new grid to code the same message you wrote in the previous slide. Exchange your newly coded message. Use it, along with the first message, to try to figure out each other’s new code. Original Input Coded Output

Compare your grid to your classmates’ new grid. In what ways are the grids the same? How are they different? For one grid, how many coded outputs are possible for one input letter? How many ways are there to decode any one letter in a coded message? Original input ABCDEFGHIJKLM Coded output Original input NOPQRSTUVRXYZ Coded output Original Input Coded Output

Use the grid at the right (page 390) to send a new two- or three-word message to your partner. Exchange and decode each other’s message. Did your partner successfully decode your message? Why or why not? Original Input Coded Output

How is the grid above different from the grid used in step 1? Code the word FUNCTION to help you answer this question. Which grid makes it easier to decode message? Which coded output letters are difficult to decode into their original letters?

Create a new coding scheme by shading squares that don’t touch each other on the grid. Make the grid so that there is exactly one output for each input. How is it similar to the grid in step 1? How is it different?

Letter-shift codes are relationships – ◦ Any relationship between two variables is called a relation. Codes that have exactly one output letter for every input letter are examples a function. ◦ The set of all input values is called the domain. ◦ The set of all output values is called the range.

Example Tell whether each table of values represents a function. Give the domain and range of each relation. InputOutput Table 1 Input101 Output125 Table 2 Input Output Table 3